a case study in the design of an optimal production ...jparsons/publications/production sharing...

Journal of Financial Economics 30 (1991) 45-67. North-Holland

A case study in the design of an optimal production sharing rule for a petroleum exploration venture*

Philip Hampson Massachusetts institute of Technology, Cambridge, MA 02139, USA

John Parsons Baruch College, C. U. N.Y, New York, NY 10010, USA

Charles Blitzer Massachusetts Institute of Technology, Cambridge, MA 02139, USA World Bank, Washington, DC 20433, USA

Received January 1989, final version received March 1990

To improve on the design of the production-sharing rule in a contract for exploration and development negotiated between a state-owned oil resources authority and a U.S. oil company, we use the Grossman and Hart (1983) principal-agent model. In the original contract, the company was granted a share of production as an incentive to maximize the net return to the authority. The optimal sharing rule we develop increases the expected return to the authority by 6% by improving the company’s incentives to choose an optimal exploration program.

1. Introduction

A key aspect of every partnership is the division of profits. Sometimes the parties agree to divide the profits on a simple 50-50 or 25-75 or similar basis. Other times the shares are contingent on the size of the realized

*This research has been supported in part by the Finance, Investment, and Contracts Program of the M.I.T. Center for Energy Policy Research and is based on Philip Hampson’s master’s thesis, ‘The Design of Optimal Production Sharing Rules for a Petroleum Exploration Venture’ (M.I.T., 1988). The research assistance of Raghuram Rajan and Sung-Hwan Shin is greatly appreciated. The final draft has benefited greatly from extensive comments and suggestions by Richard Ruback, the editor, and Kevin Murphy, the referee.

0304-405X/91/$03.500 1991-Elsevier Science Publishers B.V. All rights reserved

46 P. Hampson el al., An optimal production-sharing rule

profits, so that one partner receives a greater percentage for a marginal project than for a successful one. The finance literature gives little guidance on the choice of the sharing rule that governs a particular partnership. Presumably management’s stake should ensure that it makes the optimal investment and operating decisions. To say how large a stake is necessary for this task and to determine how it should vary with the partnership’s profits, one must carefully model the effect of the profit share on management’s decisions. Models of incentive problems have been applied in finance primarily to develop general insights about conflicts of interest between liability holders in the firm, rather than to improve the design and parameters of actual financial contracts. In this case study we show how an agency model can be used to shape and calibrate the production- or profit-sharing rule in an oil exploration partnership contract.

The contract we analyze was negotiated in mid-1986 when a state-owned oil resources authority hired a U.S. oil company to explore and develop one of its territories. Many of the important decisions involved in managing the program - how many exploratory wells to drill, which finds to develop - were to be made by the company, unmonitored by the host-country authority. To give the company an incentive to pursue an exploration and development program that would maximize the value of the host country’s resources, the contract specified that the company would share in the oil production from the territory.

The agency literature emphasizes that the parameters of the sharing rule should be tailored to the project at hand, since the strength of the incentives the sharing rule creates depends on assumptions about exploration costs and the probability of finding various amounts of oil. The terms of this contract, however, were not based on information about the territory under agree- ment. Instead, many of the terms, including the parameters for the sharing rule, were copied directly from a contract negotiated in a neighboring country. The geology of the host country, however, was strikingly different from that of its neighbor. We were therefore suspicious that the terms of the contract gave the oil company the wrong incentives and that a better set of parameters for the sharing rule could be identified, based on the actual geology of the host country.

The analysis presented here confirmed these suspicions. According to our calculations, the optimal rule would give the company a much larger share of the small discoveries than the actual contract does. Under the optimal rule, the company’s share of a discovery declines significantly as the size of the discoveries increases, rather than remaining relatively constant, as it does under the actual contract. Replacing the actual contract with the optimal sharing rule could increase the total expected return on the project by $7.8 million, or 6% of the $129 million net present value (NPV) the authority would receive under the original contract. The increased return results from

P. Hampson et al., An optimal production-sharing rule 47

better incentives to choose an optimal exploration program and the company’s financial interest in completing marginally profitable wells.

The paper proceeds as follows. In section 2 we briefly describe the negotiated contract sharing rule, and discuss how it can be redesigned to improve the incentives. One particular suggestion for an improved sharing rule is given and some alternative explanations for the shape of the original rule are discussed briefly. In section 3 we provide a numerical analysis of the incentives in the original contract and calibrate an alternative sharing rule using a principal-agent model. We compare the estimated incentive proper- ties of the two sharing rules. In section 4 we discuss in detail some of the problems raised by the numerical analysis, and section 5 concludes.

2. An analysis of the contract

Under the contract the oil company obtains exploration and development rights in a lO,OOO-square-kilometer territory owned by the authority.’ During the first three years of the contract the company agrees to drill at least one exploratory well and to produce a minimum amount of seismic data. The exploration period can be extended twice, to a total of 7; years, though the company relinquishes a portion of the territory during each extension. The contract is a hybrid profit-sharing and revenue-sharing rule. All exploration and development costs incurred by the company and the few expenses incurred by the authority are recorded in an account known as a cost recovery pool, and 40% of oil production is used to cover these costs. The remaining 60% of production is divided between the authority and the company according to a step function formula. The company receives 25% of the shared petroleum when total production lies between 0 and 25,000 barrels per day (bbl/day), 22% when total production lies between 25,000 and 50,000 bbl/day, and 20% when total production exceeds 50,000 bbl/day.

To make explicit the size of the company’s share as a function of the project NPV, we developed a simulation model based on certain assumptions about the cost of exploration and development, the size of the find, and the net sale price of oil. We then calculated the cash flows to the company and to the authority under various scenarios for exploration programs and discovery sizes.

Table 1 lists different discovery sizes, the present values of the associated oil revenues, and the company’s gross receipts, in absolute values and as a percentage of total revenues. Table 1 also reports the company’s receipts net of its expenses in developing and operating the fields, again in absolute

‘The contract studied here is a private legal instrument, and as usual in such a case, it is impossible to reveal the identity of the two parties. We gained access to the terms of the contract as a byproduct of consulting work completely unrelated to the issues raised in this paper.

Tabl

e 1

An

oil

com

pany

’s

shar

e of

to

tal

oil

reve

nues

an

d of

de

velo

pmen

t N

PV

unde

r th

e ex

plor

atio

n an

d de

velo

pmen

t co

ntra

ct

with

a

stat

e-ow

ned

oil

reso

urce

s au

thor

ity

anal

yzed

in

th

is

pape

r. Th

e co

mpa

ny’s

sh

are

is

expr

esse

d in

ab

solu

te

valu

e an

d as

a

perc

ent

of

the

tota

l re

venu

es

and

deve

lopm

ent

NPV

.

Disc

over

y si

ze q,

(bbl

/day

)

0 10

50

23

9 1,

143

5,46

7 -~

26

,141

12

5,00

0 59

7,00

0

Gro

ss

proj

ect

oil

reve

nue?

($

mill

ions

) 0

0 1

6 27

12

8 C

ompa

ny’s

gr

oss

rece

ipts

un

der

the

cont

ract

b ($

mill

ions

) 0

0 0

2 10

50

C

ompa

ny’s

re

ceip

ts

as a

per

cent

of

pro

ject

re

venu

es

_c

32.6

32

.6

32.7

37

.9

39.2

Tota

l de

velo

pmen

t N

PVd

($ m

illio

ns)

0 0

0 4

20

98

Com

pany

’s

deve

lopm

ent

NPV

” ($

mill

ions

) 0

0’

0 0

2 13

C

ompa

ny’s

de

velo

pmen

t N

PV

as a

per

cent

of

tot

al

- 10

.4

13.0

ZThe

w

ellh

ead

pric

e fo

r oi

l is

$ll/

bbl.

Dev

elop

ed

field

s ar

e op

erat

ed

for

20 y

ears

, fro

m

year

6

to y

ear

25

C

610

2,91

8 13

,954

23

9 1,

063

4,95

7 39

.2

36.4

35

.5

467

2,23

3 10

,676

59

23

1 1,

051

12.7

10

.3

9.8

lash

flo

ws

are

disc

ount

ed

at

10%

. “F

rgur

es

show

n ar

e gr

oss

rece

ipts

ea

rned

by

th

e co

mpa

ny.

Und

er

the

cont

ract

, ho

wev

er,

the

com

pany

’s

gros

s re

ceip

ts

depe

nd

in

part

on

its

docu

men

ted

expe

nses

fo

r ex

plor

atio

n an

d de

velo

pmen

t. A

ll ca

lcul

atio

ns

are

for

a sc

enar

io

in w

hich

te

n ex

plor

ator

y w

ells

ha

ve

been

dr

illed

, yi

eldi

ng

the

disc

over

y sp

ecifi

ed.

Each

ex

plor

ator

y w

ell

cost

s $8

19,7

60,

incu

rred

in

yea

r 1.

Exp

ense

s of

$5

,443

/bbl

/day

ar

e in

curr

ed

to

deve

lop

finds

of

eac

h gi

ven

size

, in

eac

h of

yea

rs

4 an

d 5.

Ope

ratin

g ex

pens

es

are

$0.7

452/

bbI.

‘No

perc

enta

ge

is c

alcu

late

d w

hen

the

deno

min

ator

w

ould

be

ze

ro.

dTot

al

proj

ect

reve

nues

fro

m

oil

sale

s le

ss

deve

lopm

ent

and

oper

atin

g ex

pens

es

incu

rred

by

the

co

mpa

ny.

Expl

orat

ion

expe

nses

ar

e co

nsid

ered

su

nk

cost

s. eT

he

com

pany

’s

gros

s re

ceip

ts

unde

r th

e co

ntra

ct

less

de

velo

pmen

t an

d op

erat

ing

expe

nses

in

curr

ed.

This

fig

ure

is

not

net

of

expl

orat

ion

expe

nses

. U

nder

th

e co

ntra

ct,

how

ever

, re

cord

ed

expe

nses

fo

r ex

plor

atio

n de

term

ine

in p

art

the

com

pany

’s

shar

e of

re

ceip

ts

from

th

e pr

ojec

t’s

oil

sale

s. ‘For

di

scov

ery

size

s be

twee

n ze

ro

and

1,00

0 bb

l/day

th

e N

PVs

of

deve

lopm

ent

are

nega

tive

and

very

sm

all.

Alth

ough

th

e co

mpa

ny

rece

ives

m

argi

nal

paym

ents

in t

hese

ca

ses,

they

do

no

t co

ver

deve

lopm

ent

expe

nses

, an

d th

eref

ore

the

com

pany

w

ill

not

deve

lop

thes

e fie

lds.


company’s percentage share of development NPV

0 1 I I I I I I I I

0 10 50 239 1,143 5,467 26,141 125,ooO 597.720

Discovery Size, q,, (MA/day)

Fig. 1. The sharing rule implicit in the actual contract. The company’s percentage share of the development NPV under the oil exploration and development contract with a state-owned oil resources authority as reported in table 1 is displayed here as a function of the size of a

discovery.

values and as a percentage of the total development net present value - i.e.. project NPV with exploration expenses excluded as sunk costs.

In absolute terms, the company’s payoff constantly increases as the discovery size increases, whereas in percentage terms the company receives the largest share of intermediate-size finds. When ten exploratory wells are drilled, the company’s share of revenues increases gradually from 32.6% to 39.2% and then declines to 35.5%. Similarly, its percentage share of the development NPV initially increases from 10% to 13% for discoveries between 1,143 and 5,467 bbl/day and then declines very slightly, to just below 10%. Fig. 1 graphs the company’s percentage share of the development NPV as a function of discovery size.

At first glance it may seem that the increasing absolute payments to the company are a sensible way to induce an optimal exploration program. The contract sharing rule gives the company ever-larger bonuses - payments in excess of development expenses - for larger discoveries. Two factors, however, suggest to us that this design is suboptimal.

First, as is now well known, an optimal sharing rule does not always have increasing bonuses. Information on the geology of the territory is an important factor in the design of the optimal rule. Increasing bonuses make sense if the company’s choice of exploration program will affect the relative probability of larger versus smaller finds. In the territory under contract, increasing the number of exploratory wells drilled increases the probability of making a discovery, but doesn’t significantly affect the relative probability of making discoveries of various sizes. Since the size of the discovery is not affected by the company’s choice of exploration program, there is no reason to make the company’s bonus contingent on discovery size. A size-contingent bonus only loads the company with risk without simultaneously conferring any incentive benefits. Instead, the optimal sharing rule would give the company a constant bonus for each discovery, regardless of its size. This fixed bonus would yield a constantly declining share of the NPV from development. At small discovery sizes the company would receive virtually all of the project value, and at very large discovery sizes its share would be extremely small.*

A second consideration leading us to believe that the original contract is suboptimal is the completion problem. Once the exploration costs are sunk and a find has been made, the company must decide whether to develop the field for commercial production. If the decision were based on the benefits shared by all parties, even very small wells of 59 bbl/day would be worth developing, but since the contract never gives the company more than 40% of the total revenues, and since the company must pay all of the development and production expenses, there is a range of discoveries larger than 59 bbl/day that should be developed but that the company will leave undevel- oped. In an optimal contract, the company would receive the full benefit of any small discoveries, to be sure it had an incentive to develop all marginally profitable wells.”

The shape of the sharing rule suggested by these two arguments is displayed in fig. 2. It contrasts sharply with the rule displayed in fig. 1.

Before proceeding to discuss in detail the parameters of the suggested optimal design, a few caveats are in order. It is possible that the original contract was written to address other agency problems than the one men- tioned here - for example, decisions about the extraction rate at any time influence not only the time profile and cost of operation, but also the total amount of oil that can be taken from a given field. In that case, the actual contract may be optimal. Alternatively, our information about the effect of

‘The relationship between the assumption about the effects of the agent’s action on the relative probability of different outcomes and the special shape of the optimal sharing rule was pointed out in Holmstrom (1979) as a rationale for deductibles in insurance contracts.

‘A comparable incentive problem was documented and analyzed by Wolfson (1985) for oil and gas partnerships in the United States.


company’s percentage share of development NPV

100

80

60

40

20

0 0 10 50 239 1,143 5,467 26,141 125,000 597,720

Discovery Size, q,, (bbliday)

Fig. 2. A suggested design for the optimal sharing rule. This design would give the company a tixed bonus payment for each find. For uneconomical finds the company would receive no payment. For marginally profitable finds the bonus would equal the full value of the discovery. For very large finds the bonus would be a very small percentage of the value of the discovery. The design is optimal for the cost and revenue assumptions reported in table 1. the discovery size probability matrix reported in table 2, and the company utility function assumptions

reported in table 5.

additional exploratory wells on the relative probabilities of discoveries of different sizes may not be accurate, in which case the structure of the optimal sharing rule might be very different and possibly close in design to the actual contract.4 There are also cases in which the most efficient structure of incentives for an optimal exploration program conflicts directly with the structure necessary to ensure an optimal completion decision. Then the structure of the optimal sharing rule would be highly contingent on the relative significance of the two effects.

These caveats, however, should not be construed as devaluing a careful modeling of the principal-agent problem. On the contrary, the sensitivity of the sharing rule design underscores the value of using an explicit model. Modeling exposes the assumptions made about the environment and clarifies the reasons for a given sharing rule. If the actual contract is optimal, the

‘For example, if increased exploration increased the relative probability of larger discoveries. the optimal sharing rule could give the company a sharply increasing share of the development NPV - see Grossman and Hart (1983, propositions 7-9).

52 P. Humpson rt al., An optimal production-sharing rule

analysis highlights the need to revise either the assumptions or the model itself. Without the application of some rigorous principal-agent model it would be virtually impossible to identify precisely the important incentive effects implicit in the probability model used or to make a compelling argument for or against any particular sharing rule. These points also highlight the genera1 conclusion that when agency problems are central, the financial contract between two parties must be carefully tailored to the cost and payoff structure of the project. In our opinion, the subject of agency theory in the field of corporate finance is precisely the exploration of the relation between the characteristics of projects and the structure of the financial contracts.

Finally, formalizing the intuition by using a detailed principal-agent model offers important advantages. For example, even if the system of increasing bonuses in the original contract makes sense as a device for giving the company strong incentives to make larger finds, the size of the bonus needs to be determined. If, for example, the company’s share increased continually to 20% for the largest discoveries instead of falling to lo%, as it does under the actual contract, the company might have an even greater incentive to pursue an exploration program that raised the probability of large discoveries. It is our objective to derive the precise values for the contract sharing rule that will induce the company to pursue an optimal exploration program and thereby maximize the value of the project to the state authority.

3. A principal-agent model

To analyze the incentives embedded in the sharing rule and to more carefully derive an optimal incentive contract, we apply the Grossman-Hart (G-H) (1983) principal-agent mode1 to our problem. The mode1 requires both a specification of how the agent’s choice of exploration program affects the probability of a discovery of any given size, and a specification of the principal’s and the agent’s preferences.

To formalize the probability relation between the exploration program and discovery size, we use a method that is consistent with approaches used in the exploration geology literature [e.g., Adelman et al. (1983)]. We specify twenty possible exploration programs, aj, j = 1,. . . , 20, representing decisions to drill one through twenty wells, respectively. The probability of making a discovery of size q given an exploration program a, is the product of(i) the probability of finding any oil, G(j), and (ii> the probability, given a find, that the discovery is of size q, H(q). A binomial distribution is used to relate the number of exploratory wells drilled to the probability of finding oil: G(j) = 1 - (1 - S)j, where 6 is called the wildcat probability. The probability that the find is a discovery of size q, H(q), is modeled as a lognormal density with parameters p and CT and is assumed independent of the number of wells


drilled. We approximate the lognormal density over discovery size with an eight-point discrete probability distribution. Each point is exactly one standard deviation apart from the next on the log scale. This process yields a 20 x 9 matrix that we refer to as the exploration-discovery matrix, in which the first element in each row denotes the probability that no discovery is made conditional on action a, and the jith element of the matrix, p,Ca,), denotes the probability of a discovery of size q, conditional on action a,. Associated with each discovery size, q,, we calculate a development NPV, r(qj), representing the value of developing the find for commercial production, exclusive of the sunk exploration costs.’ Table 2 shows part of an exploration-discovery matrix when the wildcat probability, S, is 0.4. the median discovery size, eLL, is 2,500 bbl/day, and the standard deviation of the log of discovery sizes, u, is 20%. In fig. 3 three rows from this matrix are graphed, showing the probability distributions across discovery sizes as the number of wells drilled increases.

The exploration-discovery matrix contains all the information necessary to calculate the efficient or first-best choice of an exploration program and the expected NPV for the project. The firm simply chooses the exploration program that yields the greatest expected development NPV less exploration costs. Table 3 presents the results for a number of alternative parameters describing the probability distributions. For example, if the wildcat probability, 6, is 0.4 and the median discovery size, e’l, is 2,500 bbl/day, then the first-best number of wells is nine; the cost of this exploration program is $7.38 million; the expected development NPV is $164.36 million; and the ex ante expected NPV of the exploration program is $156.98 million. The data indicate that the optimal number of exploratory wells increases as the median discovery size increases. On the other hand, as the wildcat parameter, 6. increases, the optimal number of exploratory wells drilled falls, since the probability increases that the total benefits of exploration will be captured with the first well. The marginal value of each subsequent well falls and the expense could be applied to a different territory with greater marginal return. A rough rule of thumb seems to be that doubling the wildcat probability parameter causes optimal drilling effort to fall by a factor of one-half. These first-best choices for an exploration program are unattainable in the presence of the agency costs discussed here, and our task is to find the least-cost incentive contract with which we can approach these first-best solutions.

To complete our specification of the G-H model we need to identify a utility function for the principal, the state authority, and one for the agent. the company. In the G-H model the principal’s utility function is defined over the space of possible profit levels - in our case the development NPV. The principal is typically modeled as risk-neutral, although it is possible to

‘We set ~(9,) = 0 whenever development would he a negative NPV decision

Tabl

e 2

Extra

ct

from

a

disc

over

y si

ze

prob

abili

ty

mat

rix.

Giv

en

a nu

mbe

r of

ex

plor

ator

y w

ells

dr

illed

, th

e pr

obab

ility

of

no

t fin

ding

oi

l is

cal

cula

ted

alon

g w

ith

the

prob

abili

ty

of m

akin

g fin

ds

of v

ario

us

size

s.

Also

ca

lcul

ated

is

the

co

st

of

each

ex

ulor

atio

n or

ogra

m.

the

exue

cted

va

lue

of

deve

lopi

ng

the

disc

over

ies

that

m

ight

be

mad

e,

and

the

expe

cted

va

lue

ne;

of t

he

ex&

orat

ion

cost

.

0 10

Disc

over

y si

ze

qr (

bbl/d

ay)

50

239

1,14

3 5,

467

26,1

41

125,

000

597,

720

Dev

elop

men

t N

PV

R-[

($ m

illio

ns)

Expl

orat

ion

Num

ber

of w

ildca

t co

sts

wel

ls

drill

ed

0 0

0 4

20

98

467

2,i3

3 10

,676

($

mill

ions

)

1 0.

6000

” 0.

0005

h 0.

0086

0.

0544

0.

1365

0.

1365

0.

0544

0.

0086

0.

0005

0.

8 06

6.4

065.

6 4

0.12

96

0.00

11

0.01

86

0.11

84

0.29

71

0.29

71

0.11

84

0.01

86

0.00

11

3.3

144.

5 14

1.2

8 0.

0168

0.

0013

0.

0210

0.

1337

0.

3356

0.

3356

0.

1337

0.

0210

0.

0013

6.

6 16

3.2

156.

6 12

0.

0022

0.

0013

0.

0214

0.

1357

0.

3406

0.

3406

0.

1357

0.

0214

0.

0013

9.

8 16

5.7

155.

9

Expe

cted

Ex

pect

ed

deve

lopm

ent

proj

ect

NPV

N

PV

($ m

illio

ns)

($ m

illio

ns)

“The

pr

obab

ility

of

m

akin

g no

di

scov

erie

s is

equ

al

to

1 -

(1 -

6)

‘, w

here

6

is t

he

wild

cat

prob

abili

ty

and

j is

the

nu

mbe

r of

ex

plor

ator

y w

ells

dr

illed

. Fo

r th

is

tabl

e w

e se

t th

e w

ildca

t pr

obab

ility

eq

ual

to 4

0%.

bFor

ev

ery

expl

orat

ion

prog

ram

th

e pr

obab

ility

of

mak

ing

a di

scov

ery

of a

ny

give

n si

ze

is t

he

prod

uct

of t

wo

num

bers

: (i)

the

pr

obab

ility

of

mak

ing

any

disc

over

y gi

ven

that

nu

mbe

r of

exp

lora

tory

w

ells

, S’

, an

d (ii

) th

e pr

obab

ility

of

mak

ing

a di

scov

ery

of s

ize

q,

give

n th

at

som

e di

scov

ery

is m

ade.

H

(q,).

Th

e di

strib

utio

n ov

er

disc

over

y si

zes,

H

, is

in

depe

nden

t of

th

e nu

mbe

r of

ex

plor

ator

y w

ells

dr

illed

. Fo

r th

is

tabl

e w

e sp

ecify

H

as

an

ap

prox

imat

ion

of

the

logn

orm

al

dist

ribut

ion

with

a

med

ian

disc

over

y si

ze

e fi

= 2,

500

bbl/d

ay

and

a sta

ndar

d de

viat

ion

LT =

20%

‘.

P. Hampson et al., An optimal production-sharing rule

0.80

0.60

0.40

0.20

0.00

.___ 1 wikkal well drilled ---- 4wiIdcatwellsdtilled

12 wildcat wells drilled

Discovery Size, q,, (bbllday)

Fig. 3. Discovery size probability as a function of exploration effort. Increasing the number of wells drilled lowers the probability of not making any discoveries and raises the probabilities of making discoveries of every size. The relative probability of discoveries of any given size remains constant, independent of the number of wells drilled. The graph shows a part of the matrix

reported in table 2.

incorporate risk aversion. Under the assumption of risk neutrality, the objective function for our problem is expected profit maximization. The agent is described by a utility function defined over exploration programs, a,,, and over the agent’s realized compensation, Z,, U(a,, I,> = V(Z, - II(a where Ha,) is the agent’s expense incurred in pursuing the exploration program a ;. The agent’s compensation is a vector Z = I ,, . . . , I,, whose elements are the payments made to the agent in the event of each possible realized outcome, 7r I,.‘., 7rg. The values for Zj assume that the authority reimburses the company for all development expenses, but not for exploration expenses. Therefore the payments Z, should be interpreted as the company’s share of the development NPV.

The company’s utility function must show aversion to idiosyncratic risk, Y” < 0. This requirement raises two problems: (iI the motivation for this unconventional assumption and (ii) how parameters for the utility function are to be set as well as how sensitive the results are to the choice of parameters. To retain the focus of this section on the mechanics of the model, we postpone until section 4 a discussion of these two issues. For now we simply note that the results reported here are based on the utility function ZAa,., I,> = ln(30 + Z, - II(a We have repeated the calculations for the case

56 P. Hampson et al., An optimal production-sharing rule

Table 3

First-best results. The optimal number of exploratory wells to be drilled, the cost of this exploration program, the expected value of developing the discoveries that might be made, and the expected value net of the exploration costs are calculated for a range of wildcat probability

values and median discovery sizes.

Median First-best Exploration Expected Expected discovery size e“ exploration effort cost discovery NPV project NPV

(bbl/day) (number of wells) ($ millions) ($ millions) f$ millions)

500 1,000 2,500 5,000

10,000 25,000

500 1,000 2,500 5,000

10,000 25,000

500 1,000 2,500 5,000

10,000 25,000

Wildcat probability, S = 0.02

8 6.56 16.96 10.40 I2 9.84 46.32 36.48 I7 13.94 162.29 148.35 21 17.21 418.09 400.88 26 21.31 1,089.41 1,068.10 32 26.23 3,948.29 3,922.05

Wildcat probability, 6 = 0.04

5 4.10 18.80 14.70 7 5.74 48.35 42.61 9 7.38 164.36 156.98

11 9.02 420.45 411.43 13 10.66 1,091.30 1,080.64 16 13.12 3,950.31 3,937.19


3 2.46 19.08 16.62 4 3.28 47.38 44.10 6 4.92 165.35 160.43 7 5.74 421.29 415.55 8 6.56 1,091.9X 1,085.42 9 7.38 3,950.38 3,943.oo

of exponential utility and the qualitative shape of the derived sharing rule does not change, although the precise values calculated vary slightly.

With the actual contract sharing rule, I* = IF,. . . , It, the company chooses the exploration program that maximizes its utility:

aA maximizes i [Pi(aj)u(aj? 'A)]'

U,E(U,,...3a20) i=l

The exploration program the contract induces the company to choose and the expected returns to the authority are displayed in table 4. For the scenario in which the median discovery size, e@, is 2,500 bbl/day and the wildcat probability, 6, is 0.4, the contract sharing rule achieves an expected profit of $129.05 million for the authority, or roughly 82% of the first-best expected level. Although some of the lost profit may be the result of the poor


Table 4

Expected results of the contract described in table 1. For a range of wildcat probability values and median discovery sizes and given the terms of the actual contract, we calculate the number of exploratory wells drilled that will maximize the return to the company. Also calculated is the cost of these exploration programs, the expected payment made to the company under the

contract, and the net value remaining for the authority after it makes the payment.

Median Exploration Expected payment’ Expected return discovery size ep Exploration efforta cost to the company to the authority

(bbl/day) (number of wells) ($ millions) ($ millions) ($ millions) ~ ~~

500 1,000 2,500 5,000

10,000 25,000


0 0.00 0.00 0.00 0 0.00 0.00 0.00 6 4.92 12.55 110.00 9 7.38 40.3 1 325.04

12 9.84 106.28 911.33 16 13.12 383.54 3.456.65

Wildcat probability, S = 0.04

500 0 0.00 0.00 0.00 1,000 2 1.64 2.93 28.90 2,500 4 3.28 15.46 129.05 5,000 6 4.92 42.55 359.75

10,000 7 5.74 109.58 952.53 25,000 8 6.56 385.93 3A99.12


500 0 0.00 0.00 0.00 1,000 2 1.64 3.84 37.94 2,500 3 2.46 16.33 139.07 5,000 4 3.28 43.02 368.16

10,000 5 4.10 110.86 970.65 25,000 6 4.92 390.31 3,544.92

~- .______ aThe number of exploratory wells that maximizes the company’s utility given the terms of the

actual contract: UfI, a) = log(30 + I - D(a)), where I is the payment received by the company and D(a) is the cost of the chosen exploration program.

bThe expected payment is calculated as the company’s share of the project revenues less its share of the development expenses. The company’s expenses for the exploration program are not included and ex ante the company must expect the payment to cover them.

contract design, some of it is an unavoidable cost of giving the company the right incentives. The only way to determine the unavoidable portion is to use our model to set parameters for the optimal sharing rule, and it is to this problem that we now turn.

The optimal or second-best sharing rule, Z = IT”, . . . , Z9SB, is derived using a pair of mathematical programming problems. With the first program, we choose, for each number of exploratory wells drilled, (i) the least-cost incentive contract, Z*(aj), for which (ii) it is in the company’s interest to accept the contract as against the best available alternative, and (iii) it is in


the company’s interest to choose the specified number of exploratory wells as against any other number of wells. That is,

VU,, Z*(Uj) si [Z:(Uj),...,z:(aj)]rnntrt7i~es C [Pi("j)l,],

I) -9 i=l

such that 9

C [ Pi(“,)U(aj~ I,)] 2 ‘min, i=l

and such that

Vu, # uj, 5 [ Pi(“j)U(uj~zi)] L i$l [Pi(“k)U(ak~ ‘i)l. i=l

For most of the results presented in the paper we have added the constraint that the payment to the company be both nonnegative and bounded above by the total development NPV, 0 s Z, I ri. With the second program, we find the number of exploratory wells that gives the authority the greatest development NPV net of the payments it must make to the company. That is,

uSB maxhmizes i [Pi(aj)[TL-z?(uj)]]’

i=l

In table 5 the optimal sharing rule is displayed for the case in which the median discovery size is 2,500 bbl/day and the wildcat probability is 0.4 - the

Table 5

The optimal sharing rule. The payment - contingent on discovery size - to be made to the company under the calculated optimal contract is displayed in absolute value and as a percent-

age of the development NPV for the discovery.

Discovery size qr (bbl/day)

0 10 50 239 1,143 5,467 26,141 125,000 597,720

Development NPV rr, ($ millions)

0 0 0 4 20 98 467 2,233 10,676

Payment to the companya in $ millions, I, 0 0 0 4 20 30 30 30 30 as a % share, 1,/r, 0 0 0 100 100 30.6 6.4 1.3 0.3

aPayments are derived under the assumption that the wildcat probability of making a discovery is 40%, that the median discovery size is 2,500 bbl/day, and that the standard deviation on the log of discovery size is 20%. Also assumed is the company utility function U(I, a) = log(30 + I -D(a)), where I is the payment received by the company and D(a) is the cost of the chosen exploration program. The payment is net of all development expenses incurred, but not net of the expenses incurred for exploration. Ex ante the company must expect the payment to cover these expenses.


same sharing rule was graphed in fig. 2. A careful look at the payments made under this rule will help us understand the incentive problem for this project. Of course when no discovery is made, the company receives no payment. The company also receives a zero payment for discoveries of 10 or 50 bbl/day, i.e., for discoveries that should not be developed. For discovery sizes 239 and 1,143 bbl/day, the company receives a payment equal to the full development NPV: our ceiling constraint on the payment is binding for these two events. For discovery sizes ranging from 5,467 to 597,720 bbl/day the company receives payments that are approximately invariant at $30 million. The company’s share of the development NPV falls from 100% for discoveries of 239 and 1,143 bbl/day to 30.6% for a discovery of 5,467 bbl/day and to 0.3% for the largest discovery.

There is a clear rationale for this shape of the optimal sharing rule. The efficient agency contract is to pay the company a fixed bonus whenever a commercial discovery is made. Our sharing rule has this characteristic, except that at low discovery sizes the constraint that the incentive payment cannot exceed the total NPV is binding. The optimal& of a fixed bonus is a consequence of two earlier assumptions. First, the authority is risk-neutral while the company is risk-averse. Therefore it is optimal for the authority to bear all of the risk except when making the company’s income contingent on the development NPV provides an incentive for more exploration. Second, the probability matrix relating exploration levels to outcomes has been specified so that the probable size of any discovery is independent of the exploration effort. Therefore, the absolute size of the discovery gives the authority no information about the company’s choice of an exploration program, so there are no incentive benefits to making the company’s income contingent on discovery size. Instead, it is optimal to give the company a fixed bonus - approximately $29.97 million - whenever a commercial discovery of any size is made.6

In table 6 we display the optimal sharing rules for alternative parameters defining the exploration-discovery matrix. The first two columns list these parameters. The number of exploratory wells to be drilled in the second-best exploration program is shown in the third column. The remaining nine columns list the payments, net of development costs, to be made to the company for each of the eight possible discovery sizes as well as the event of

‘In addition to the argument of Holmstrom (1979) referred to earlier, the proof used for proposition 9 of Grossman and Hart (1983) helps explain the reason for this special shape of the sharing rule. In the proof they describe certain situations in which the expression [1/1/‘(1, + ,) - l/V’(1,)] must be proportional to the expression ((?r,+,(a,)/?r,+ l(ase)) - (T,(u,)/T~(u~~)). Although all of the conditions of the proposition do not obtain in our case, over one range of outcomes - i = 2,. ,8 - the equation used in the proof is also determinant for our problem. Since over the range of i = 2,. . ,8 this second expression is constant, so too must the first expression be constant, and this can be true only if I, = I, + , for i = 2.. (8, which is the case for our sharing rule.

Tabl

e 6

g

Opt

imal

sh

arin

g ru

les

with

di

ffer

ent

para

met

ers.

The

paym

ent

- co

ntin

gent

on

di

scov

ery

size

-

to

be

mad

e to

th

e co

mpa

ny

unde

r th

e ca

lcul

ated

op

timal

co

ntra

ct

is d

ispl

ayed

in

abs

olut

e va

lue

for

a ra

nge

of w

ildca

t pr

obab

ility

va

lues

an

d m

edia

n di

scov

ery

size

s.

Also

di

spla

yed

is t

he

num

ber

of

expl

orat

ory

wel

ls

the

com

pany

w

ould

fin

d it

optim

al

to d

rill

unde

r th

e co

ntra

ct,

asn.

Med

ian

disc

over

y si

ze e

F (b

bl/d

ay)

Wild

cat

prob

abili

ty

6 (%

)

Cho

sen

Inde

x of

the

di

scov

ery

size

i

num

ber

of

, ?

1 n

< lz

7

P

9 ex

plor

ator

y w

ells

aS

B

500

0.2

4 50

0 0.

4 3

500

0.6

2

l,W

1,00

0 1,

000

2,50

0 2,

500

2,50

0

5,00

0 5,

000

5,00

0

10,0

00

10,0

00

10,0

00

25,0

00

25,0

00

25,0

00

0.2

0.4

0.6

0.2

0.4

0.6

6 4 2 9 6 3

0.2

12

0.4

6 0.

6 5

0.2

0.4

0.6

0.2

0.4

0.6

14 8 4 17 8 5

I L

.7

-t

J ”

”

Paym

ent

to t

he

com

pany

co

ntin

gent

on

di

scov

ery

size

2,

s

15.1

7 2

8.28

$

3.80

2

0 0

0 0.

44

4.80

13

.56

13.5

4 13

.55

0 0

0 0.

44

4.80

7.

87

7.86

7.

85

0 0

0 0.

44

3.93

3.

93

3.93

3.

95

0 0

0 1.

57

8.95

18

.57

18.5

5 18

.54

0 0

0 1.

57

8.95

11

.40

11.4

0 11

.42

0 0

0 1.

57

3.73

3.

73

3.73

3.

75

0 0

0 4.

21

20.4

2 29

.03

29.0

8 29

.13

0 0

0 4.

21

20.4

2 29

.99

29.9

8 29

.71

0 0

0 4.

21

8.65

8.

65

8.66

8.

60

0 0

0.28

6.

94

38.1

1 51

.43

51.3

6 51

.67

0 0

0.28

6.

94

25.0

2 25

.02

25.0

0 24

.77

0 0

0.28

6.

94

20.1

6 20

.15

20.1

7 20

.27

0 0

0.96

11

.27

65.9

3 65

.92

66.0

9 67

.03

0 0

0.96

11

.27

61.8

7 61

.99

62.0

2 62

.70

0 0

0.96

11

.27

19.3

3 19

.33

19.3

3 19

.29

0 0

2.32

21

.40

111.

52

116.

78

122.

24

104.

44

0 0

2.32

21

.40

57.3

1 57

.33

57.5

2 57

.33

0 0

2.32

21

.40

41.8

7 41

.87

41.8

5 41

.94

20.7

3 R

10.8

8 a

3.60

; rZ

. 33

.62

9 29

.56

!L

9.65

2 k

48.0

3 E

24.8

7 =.

18

.90

$_

51.0

9 %

17

3.52

” 8.

19

.30

2

5465

.30

6 16

7.36

41

.65

“Val

ues

for

the

larg

est

paym

ent

to

the

com

pany

ar

e so

met

imes

ex

trem

e -

for

exam

ple,

w

hen

e@ =

10,

000

and

S =

0.4

or

whe

n e”

=

25,0

00

and

6 =

0.2.

Th

is

occu

rs

beca

use

(i)

the

prob

abili

ty

of

this

ev

ent

is v

ery

smal

l, an

d (ii

) ro

undi

ng

is d

one

in

the

proc

ess

of

cons

truct

ing

the

disc

rete

pr

obab

ility

m

atrix

. Th

e co

nstru

ctio

n of

the

op

timal

sh

arin

g ru

le

play

s he

avily

on

th

e sm

all

chan

ges

in r

elat

ive

prob

abili

ties

acro

ss

even

ts.

Sinc

e th

e pr

obab

ility

of

the

se

even

ts

is s

o sm

all,

how

ever

, th

e la

rge

num

bers

ne

ver

signi

fican

tly

chan

ge

the

tota

l ex

pect

ed

paym

ent

to

the

com

pany

.


Table 7

Second-best results. For a range of wildcat probability values and median discovery sizes and given the terms of the calculated optimal contract, we calculate the number of exploratory wells drilled that will maximize the return to the company. Also calculated is the cost of these exploration programs, the expected payment made to the company under the contract, and the

net value remaining for the authority after it makes the payment.

Median discovery size e@

(bbl/day) ~_

500 1,000 2,500 5,000

10,000 25,000

500 I .ooo 2,500 5,000

10,000 2s.000

500 1,000 2,500 5,000

10,000 25,000

Second-best Exploration Expected payment exploration effort cost to the company (number of wells) ($ millions) ($ millions)


4 3.28 5.01 6 4.92 9.26 9 7.38 19.12

12 9.84 36.93 14 11.48 54.52 I7 13.94 98.23


3 2.46 4.42 4 3.28 7.81 6 4.92 21.48 6 4.92 20.96 8 6.56 52.83 8 6.56 57.32


2 1.64 2.83 2 1.64 2.82 3 2.46 7.35 4 3.28 17.45 4 3.28 17.40 5 4.10 37.75

Expected return to the authority

($ millions)

7.03 27.44

124.69 356.03 989.81

3.757.80

11.57 35.49

136.81 381.34

1,021.60 3.835.80

14.29 38.96

148.09 393.83

1,047.70 3.874.90

no discovery. Since the absolute sizes of the eight discoveries are different for each set of probability parameters, the columns are indexed only by the ordinal size of the discovery.

Table 7 summarizes the project results under the second-best sharing rule for the same set of alternative parameters. For the scenario in which the median discovery size, e”‘, is 2,500 bbl/day and the wildcat probability, 6, is 0.4, the optimal sharing rule yields an expected profit of $136.81 million for the authority, or roughly 87% of the first-best expected level. The 13% difference is the cost the principal must pay to give the agent the appropriate incentives. Incentive costs - measured as a fraction of the first-best profit - decrease as median discovery size and the wildcat probability increase. The incentive cost is 32% of first-best expected profits for a median discovery size of 500 bbl/day with a wildcat probability of 0.2 and decreases to 4% of first-best profits for a median discovery size of 25,000 bbl/day.

62 P. Hampson et al., An optimal productlowsharing rule

Table 8

A comparison of the results of the optimal and actual contracts.”

Cost of the Expected Expected Number exploration program payment to return to of wells chosen the company the authority drilled ($ millions) ($ millions) ($ millions)

First-best 9 7.38 7.38 156.98

Actual contract 4 3.28 IS.46 129.05

Optimal contract/ second-best 6 4.92 21.48 136.81

“Figures are taken from tables 4, 5, and 7 for the case in which the wildcat probability is 40% and the median discovery size is 2,500 bbl/day.

Measured in absolute value, however, the incentive cost increases from $3.37 million to $164.25 million. As we change the wildcat probability from 0.2 to 0.6, keeping the median discovery size constant at 500 bbl/day, the incentive cost falls from 32% of first-best expected profits to 15% - from $3.37 million to $2.33 million.

It is now possible to compare the results of the optimal contract design with the results of the actual contract sharing rule - see tables 4 and 7. One can see that the ex ante expected profits to the authority are significantly lower under the actual contract. For example, for the scenario in which the median discovery size, eP, is 500 bbl/day, regardless of the wildcat probability the project is worthless under the actual contract, whereas under the optimal sharing rule the authority realizes between $7 and $14 million profit. For the scenario in which the median discovery size is 2,500 bbl/day and the wildcat probability is 0.4, the actual contract gives the authority $7.76 million less in expected profit than the optimal sharing rule, a loss of 5.7% of the total project value. For larger values of the median discovery size and the wildcat parameters, the dollar loss with the actual contract increases significantly - up to $336 million. A summary comparison of the results for the first-best rule, the actual contract, and the optimal contract is given in table 8.

The actual contract is less favorable to the authority than the optimal sharing rule primarily because incentives in the actual contract are expensive. In the actual contract the size of the payment to the company is highly correlated with the size of the discovery. But given that a find is made, the absolute size of the discovery is an exogenous random variable unrelated to the company’s choice of an exploration program, so the randomness in the payment does not increase the company’s incentive to expand its exploration program. Moreover, the average payment to the company must be increased to compensate for the risk the company bears under the actual contract. In


the optimal sharing rule the company receives nothing when no discovery is made, and a frxed bonus when a discovery of any size is made. Since a discovery conveys some information about the extent of the exploration program, the bonus increases the company’s incentive to expand the exploration program, but since the bonus doesn’t vary with the size of the discovery, the risk borne by the company is minimal.

4. Discussion

The explicit use of a utility function for the agent/company in this paper raises an interesting problem in the application of some agency models to corporate finance. It is often assumed that managers of a firm choose projects as if the firm were neutral toward idiosyncratic risk. This assumption corre- sponds correctly with the decision rule that results if shareholders have diversified their investments and capital markets are perfect. If the management of the corporation, or the agent, is neutral toward the project’s idiosyncratic risk, then, as Harris and Raviv (1979) show, the efficient incentive contract takes on a trivial form. The principal, in this case the authority, should simply sell the rights to exploration and development to the agent, the company, for a fixed fee and allow the company to bear all of the risk. The actual contract in the case under study does not fit this description. Thousands of comparable contracts are negotiated with similarly complex designs that do not correspond to the predictions of models based on the agent’s risk neutrality. We believe that in many cases these modeling assumptions and the incentive contract they predict do not correspond very closely to the real context.

Our assumption that the company is to some degree averse to the idiosyncratic risks of the project can be justified in the framework of an appropriate equilibrium model of a capital market with imperfect information. The assumption of risk neutrality is a strong one that makes sense only in a model of relatively perfect information and frictionless capital markets. If, however, the management of each firm possesses private information about the projects in which it is investing and about its own management of these projects, we believe the consequent imperfections in the financial markets lead the firm rationally to behave as if it were averse to the idiosyncratic risk associated with each particular project. Holmstrom (1982) makes a similar suggestion. The environment we have in mind could even incorporate indi- vidual investors who are risk-neutral and who hold diversified portfolios. Rather than focus on modeling the environment that might generate this risk aversion, we choose in this paper to take it for granted and focus instead on the interesting problem of how to design and calibrate an optimal sharing rule between the authority and the company in this environment.


Of course, having decided to recognize the risk aversion that probably characterizes most corporate activities, we then face the problem of specify- ing the utility function. This problem has not been seriously explored in the finance literature, so there is dangerously little basis on which to specify the model. The results presented here assume that the agent’s utility function is log(30 + I - D(a)>: we began with this specification because it describes a project manager with a limited exploration budget of $30 million, a budget more than adequate to cover any of the possible first-best exploration programs, and in some cases much more. Fortunately in our case the key result of the analysis is unaffected by the parameters of the utility function: although the exact values of the payments made to the agent are obviously subject to the choice of utility function, the shape of the optimal sharing rule does not change. The actual contract cannot be made optimal by altering the parameters of the agent’s utility function. It is possible to prove that under all parameter choices the optimal sharing rule involves a zero payment for small finds and a fixed bonus for large finds, with the size of the bonus unaffected by the size of the find. The motivation for this shape was provided in section 2 above. Of course, in the application of this model to other cases, the parameters of the agent’s utility function may be significant in determining the optimal shape of the sharing rule. This issue therefore requires attention in the future.

We have modeled the problem as if the principal were risk-neutral. This assumption is made in the theoretical literature primarily because risk aversion is not necessary for interesting results, and the problem and its solution are much clearer if the principal is risk-neutral. The Grossman-Hart model can easily be adapted for a risk-averse principal. Many arguments can be made for modeling the principal in any given case as risk-averse or risk-neutral, but as Holmstriim (1982) points out, the principal’s aversion to the idiosyncratic risk of the project does not immediately follow from the capital-market imperfections that make the agent risk-averse.

Instead of debating whether the principal should be modeled as averse to the idiosyncratic risk, we merely note that a change in this assumption has important consequences for the design of the optimal sharing rule and the comparison of the actual contract with it. Table 9 shows the optimal sharing rule when the authority shows the same risk aversion as the company. The payments to the company increase very moderately with discovery size, so the company shares a portion of this exogenous risk. The company’s share of the development NPV declines significantly as the discovery size increases, moving from 100% to 0.2%. The structure of the sharing rule is therefore very similar to the one displayed in table 5, the case of the rise-neutral principal. The authority’s risk aversion leads to a sharing rule in which the authority also receives a portion of the value of small discoveries: the company’s share of a discovery of 1,143 bbl/day is only 38.2% instead of


Table 9

The optimal sharing rule with a risk-averse principal. The payment - contingent on discovery size - to be made to the company under the calculated optimal contract under the assumption that the authority is also risk-averse is displayed in absolute value and as a percentage of the NPV of developing the field. The contract simultaneously provides incentives to the company

and optimally shares the risk between the company and the authority.

Discovery size qr (bbl/day)

0 10 50 239 1,143 5,467 26,141 125,000 597.720 -____-.

Development NPV r, ($ millions)

0 0 0 4 20 9x 467 2,233 10.676

Payment to the company” in $ millions, I, 0 0 0 4 8 23 44 37 27 as a o/o share, I,/rr, 0 0 0 100 38.2 23.2 9.4 1.7 0.2

aPayments are derived under the assumption that the wildcat probability of making a discovery is 40%. that the median discovery size is 2,500 bbl/day, and that the standard deviation on the log of discovery size is 20%. Also assumed is the company utility function U(I, a) = logC30 + I-D(a)), where I is the payment received by the company and D(a) is the cost of the chosen exploration program. The payment is net of all development expenses incurred, but not net of the expenses incurred for exploration. Ex ante the company must expect the payment to cover these expenses. The authority is assumed to be risk-averse with a utility function log(30 + T - I).

100% and its share of a discovery of 5,467 bbl/day is 23.2% instead of 30.6%,. Of course, the company must receive a slightly increased share in other events to compensate for these reductions. Although the exact values for the payment to the company do change, the general structure of the optimal sharing rule remains as we have described it. In particular, under all cases (i) it is important that the company receive a large fraction of the development NPV when the discovery is small, and (ii) although the absolute dollar value of the payment to the company may be increasing, it is always a sharply decreasing share of the development NPV. This sharing rule remains in sharp contrast to the one implicit in the actual contract, and the expected returns to the authority remain far greater.

Another important assumption implicit in our use of the Grossman-Hart model is that the company makes an essentially one-shot decision. In one respect this assumption is reasonable, since most of the uncertainty in an oil exploration venture is resolved in a relatively short period. At the end of that time a decision is made whether to develop the territory for commercial production. After this point the operating wells can in some cases be properly viewed as cash cows. In some important respects, though, the exploration decision is not a one-shot problem. For example, one could view the exploration program as involving a Bayesian updating process in which the results from the first wells drilled provide information about whether

66 P. Humpson et al., An optimal production-sharing rule

additional wells should be drilled or whether the program should be aban- doned. The number of wells drilled is then the result of a dynamic stochastic program and not a simple ex ante choice variable, as in our model. Simplify- ing the decision problem as we have done obviously sacrifices some accuracy and some interesting components of the problem, but we think the value of our results justifies this compromise. An interesting research problem would be to analyze the company’s choice of exploration program as a dynamic agency problem and to consider the actual contract as an attempt to provide the optimal incentives for this type of problem.

As a final caveat, we treat the oil price as if it were known ex ante and constant over the life of the contract. To allow for an uncertain future oil price, we must modify several aspects of our analysis. First, we must be more careful about the exact terms in which the contract is written. The actual contract specifies shares in quantity of oil that each party will receive: hence, when the oil price changes, the absolute value of the payments to the company will change even though its share of the oil produced does not. On the other hand, we have solved for the optimal sharing rule in terms of the absolute value of the dollar payments that should be made to the company in the event of different discovery sizes. When the oil price changes, these payments should remain fixed; they will represent changing fractions of the development NPV. Given that oil prices are variable, in which terms should the optimal contract be written ? The company’s choice of an exploration program affects the probability of a discovery, and not the present value of that discovery. Therefore the company should receive an incentive payment whenever it makes a discovery, and the payment should not depend on the realized value of the discovery and thereby on the price of oil. This conclusion is modified slightly when the principal is risk-averse, since the risk implicit in the variable price then needs to be shared between the two parties - although not for incentive purposes. Solving for the optimal sharing rule in the face of an uncertain oil price gives results virtually identical to those presented for a fixed oil price. There may be other reasons to write the contract in terms of shares of oil, but from the standpoint of incentives for exploration in a model such as ours this is not the preferred form.

5. Conclusion

We analyze a classic capital budgeting case in the form of an oil exploration project, and we value in the classic manner the contract written between the owner of the territory and the manager of the exploration and development program. We then analyze the same contract using a principal-agent model, quantifying the incentives given to the manager of the exploration and development program, and valuing the project again in light of these incentives. We solve for an optimal incentive contract and contrast it


with the actual one. Our results demonstrate that it is possible to improve significantly on the design of the original contract, using an existing principal-agent model. In the course of this demonstration we highlight several difficulties in the practical application of the model.

References

Adelman, Morris A., 1986, The competitive floor to world oil prices, Energy Journal 7. 9-35. Adelman, Morris A., John C. Houghton, Gordon Kaufman, and Martin Zimmerman, 1983,

Energy resources in an uncertain future (Ballinger, Cambridge, MA). Grossman, Sanford J. and Oliver D. Hart, 1983, An analysis of the principal-agent problem.

Econometrica 51, 7-45. Harris, Milton and Artur Raviv, 1979, Optimal incentive contracts with imperfect information.

Journal of Economic Theory 20, 231-259. Holmstrom, Bengt, 1979, Moral hazard and observability, Bell Journal of Economics 10, 74-91. Holmstrom, Bengt, 1982, Moral hazard in teams, Bell Journal of Economics 13, 324-340. Wolfson, Mark A., 1985, Empirical evidence of incentive problems and their mitigation in oil

and gas tax shelter programs, in: J.W. Pratt and R.J. Zeckhauser, eds., Principals and agents: The structure of business (Harvard Business School Press, Boston, MA) 101-125.

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